Necessary and sufficient condition for stability of generalized expectation value

A class of generalized definitions of expectation value is often employed in nonequilibrium statistical mechanics for complex systems. Here, the necessary and sufficient condition is presented for such a class to be stable under small deformations of a given arbitrary probability distribution.

Given a probability distribution p i { } i =1, 2, ..., W , i.e., 0 ! p i ! 1 ( i = 1, 2, ..., W ) and p i i =1 W ! = 1 , the ordinary expectation value of a quantity Q of a system under consideration is defined by where W is the total number of accessible states and is enormously large in statistical mechanics, typically being 2 10 23 . In the field of generalized statistical mechanics for complex systems, on the other hand, discussions are often made about altering this definition. Among others, the so-called "escort average" is widely employed in the field of generalized statistical mechanics [1][2][3]. It is defined as follows: where P i (! ) stands for the escort probability distribution [4] given by with a nonnegative function ! . In the special case when ! (x) = x , Q ! is reduced to the ordinary expectation value mentioned above.
Consider measurements of a certain quantity of a system to obtain information about the probability distribution. Repeated measurements should be performed on the system, which is identically prepared each time. Suppose that two probability distributions, Here is the l 1 -norm describing the distance between these two probability distributions. One might consider norms of other kinds, but what is physically relevant to discrete systems is the present l 1 -norm [5]. Equation (3) is analogous to Lesche's stability condition on entropic functionals [5], which has recently been revisited in the literature [6][7][8][9][10][11] (note that the discussion in Ref. [8] is corrected in Ref. [9]). This concept of stability is actually equivalent to that of uniform continuity.
In recent papers [12,13], it has been shown that the generalized expectation value in equation (1) with a specific class, ! (x) = x q ( q > 0 ), (the associated expectation value being termed the q-expectation value), is not stable unless q = 1 . This result need the q-expectation-value formalism of nonextensive statistical mechanics [1,2] be reconsidered. In addition, the result is supported by Boltzmann-like kinetic theory in an independent manner [14].
Here, it seems appropriate to make some comments on the latest situation of the problems concerning stabilities of entropic functionals and generalized expectation values. The authors of Refs. [15,16] have presented discussions which aim to rescue the q-expectation values from the difficulties of their instability pointed out in Ref. [12].
Those authors insist that the q-expectation values can be stable in both the finite-W and continuous cases. Such possibilities are however fully refuted by the work in Ref. [13] both physically and mathematically, and the controversy seems to have been terminated with that work. The case of the continuous variables was further been carefully examined in a recent paper [17], where the so-called Tsallis q-entropies [1,2] do not have the continuous limit in consistency with the physical principles such as the thermodynamic laws (see also [18,19]). These controversies have led the researchers to give up the traditional form of nonextensive statistical mechanics based on the q-entropies and q-expectation values and to examine other entropic functionals combined with the ordinary definition of expectation values [20] (see also Refs. [21,22]). Thus, it seems that nonextensive statistical mechanics has to be fully reexamined, theoretically.
In this paper, we present the necessary and sufficient condition for Q ! [ p] in equation (1) Proof. First, assume that lim ! (x) / x does not vanish because of the condition ! (x) = 0 " x = 0 . Therefore, there Putting c = min {a / 2, b} , we have Consequently, for an arbitrarily large W and an arbitrary probability distribution From the mean value theorem, it follows that where ! '(x) is the derivative of ! (x) with respect to x. For ! > 0 , we put where Therefore, Q ! [ p] is stable.
On the other hand, suppose that lim " (x) / x = # . Below, we shall examine these cases separately.
(i) Consider the following deformation: which are normalized and satisfy p ! p ' 1 = " . We have Difference of the expectation values is calculated as follows: which are also normalized and satisfy p ! p ' 1 = " . We have Difference of the expectation values is calculated as follows: since lim In the above proof, we have employed the specific deformations of the probability distributions as the counterexamples, which are considered in Ref. [5]. It is pointed out in Ref. [13] that these deformed distributions may experimentally be generated.
Finally, we mention a couple of simple stable examples.
Example 2: which yields a stable generalized expectation value, if and only if ! = 1.
In conclusion, we have considered a class of generalized definitions of expectation value that are often employed in nonequilibrium statistical mechanics for complex systems, and have presented the necessary and sufficient condition for such a class to be stable under small deformations of a given arbitrary probability distribution.